The theory of finite groups is of central importance in mathematics, and finds wide applications in all the physical sciences and elsewhere. In essence it is the deep study of symmetry in all its innumerable manifestations, and so has applications in all situations where symmetry occurs. The building blocks of finite groups are the ‘simple ’ groups, analogous to the prime numbers in number theory, which are socalled because they cannot be broken down into smaller pieces. The study of ‘simple ’ groups, however, just like the study of prime numbers, turns out to be not simple at all. The completion around 1980 of the massive world-wide project for complete classification of the finite simple groups (see, for example, [31]), revealed that they mostly fall into various reasonably well understood families, with precisely twenty-six exceptions, known as the ‘sporadic ’ simple groups. These range